Upside down
These are level 3 number, geometry and statistics problems from the Figure It Out series. They are focused on finding fractions of regions, solving addition problems involving money, exploring rational symmetry, and investigating bar and pie graphs. A PDF of the student activity is included.
About this resource
Figure It Out is a series of 80 books published between 1999 and 2009 to support teaching and learning in New Zealand classrooms.
This resource provides the teachers' notes and answers for one activity from the Figure It Out series. A printable PDF of the student activity can be downloaded from the materials that come with this resource.
Specific learning outcomes:
- Find fractions of regions (Problem 1).
- Solve addition problems involving money (Problem 2).
- Explore rotational symmetry (Problem 3).
- Investigate bar and pie graphs (Problem 4).
Upside down
Achievement objectives
GM3-6: Describe the transformations (reflection, rotation, translation, or enlargement) that have mapped one object onto another.
NA3-1: Use a range of additive and simple multiplicative strategies with whole numbers, fractions, decimals, and percentages.
S3-1: Conduct investigations using the statistical enquiry cycle: gathering, sorting, and displaying multivariate category and wholenumber data and simple time-series data to answer questions; identifying patterns and trends in context, within and between data sets;communicating findings, using data displays.
Required materials
- Figure It Out, Level 3, Problem Solving, "Upside Down", page 8
See Materials that come with this resource to download:
- Upside down activity (.pdf)
Activity
Encourage students to visualise and use reasoning before they measure the rectangles, though this measuring can be used to confirm their thinking. They will have to use fractional knowledge in this problem.
This could be confirmed by fitting the small rectangle into the large one 16 times, as shown.
Using similar reasoning, a quarter of the small rectangle must be 1/4 of 1/16, which is 1/64 (one sixty fourth).
Sarah will receive $2 – $1.20 = 80c in change. Students will need to look for combinations of four coins that add to 80 cents. The results can be organised into a table to eliminate combinations that don’t work and to avoid duplication.
Coins |
Number of coins |
|||
|---|---|---|---|---|
5c |
10c |
20c |
50c |
|
2 |
1 |
1 |
4 |
|
3 |
1 |
4 |
||
|
|
4
|
|
4 |
Systematically working through by starting with combinations involving 5 cent coins will find all the possibilities.
This problem uses rotational symmetry. All the numbers used in the expression are either:
- images of themselves after a half turn, for example:
or
- mirror images of another number.
The matching numbers are shown below.
The answers are the same because the same numbers are added.
One way to confirm that these graphs are showing the same data is to imagine how the bar graph could be transformed into a pie graph.
Another method is to apply fractional knowledge to the numbers involved. The bars have heights of 4, 6, and 2 respectively, so there are 4 + 6 + 2 = 12 items of data. The red bar has height 4, so it is 4/12 or 1/3 of the number of the data items. The blue has height 6, so it is 6/12 or 1/2 of the data items. The pie chart shows fractions of 1/3, 1/2, and 1/6, so the representations match.
1.
a. 1/16 (0.0625)
b. 1/64
2.
4 x 20c; 50c, 10c, 10c, and 10c; or 50c, 20c, 5c, and 5c
3.
a. 3 481
b. You get the same answer.
c. When the book is turned upside down, you are looking at exactly the same set of numbers being added together as before because 8 and 1 have half-turn symmetry, 9 is a half-turn image of 6, and 2 is a half-turn image of 5.
4.
Yes. The first bar is showing 4 out of 12 units, the second is showing 6 out of 12, and the third 2 out of 12. Expressed as fractions, this is 4/12 = 2/6, 6/12 = 3/6, and 2/12 = 1/6. These are the same fractions shown by the pie graph.
Students’ stories will vary.
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